The Cross-Sectional Shapes of Longitudinal Hydraulic Fractures

1984 ◽  
Vol 106 (4) ◽  
pp. 554-561 ◽  
Author(s):  
D. Segalman
Keyword(s):  
Mathematical Formulation ◽  
High Stress ◽  
In Situ Stress ◽  
Hydraulic Fractures ◽  
Stress Distributions ◽  
Type Problem ◽  
Cross Sectional ◽  
The Cross ◽  
Sectional Shape

A mathematical formulation has been developed for calculating the cross-sectional shape of hydraulic fractures. This formulation treats the problem as a free-boundary-type problem and is modeled after mathematical formulations developed for contact and lubrication problems. Numerical solution of the resulting equations has been used to address problems involving particularly difficult in-situ stress distributions, including problems in which the fracture breaks through high-stress barriers. The technique is illustrated on two example problems.

scholarly journals Apparent Toughness Anisotropy Induced by “Roughness” of In-Situ Stress: A Mechanism That Hinders Vertical Growth of Hydraulic Fractures and Its Simplified Modeling

SPE Journal ◽  
2019 ◽  
Vol 24 (05) ◽  
pp. 2148-2162 ◽  
Author(s):  
Pengcheng Fu ◽  
Jixiang Huang ◽  
Randolph R. Settgast ◽  
Joseph P. Morris ◽  
Frederick J. Ryerson
Keyword(s):  
Hydraulic Fracture ◽  
High Stress ◽  
In Situ Stress ◽  
Height Growth ◽  
Hydraulic Fractures ◽  
Simple Relationship ◽  
Vertical Growth ◽  
Stress Profiles ◽  
Fracture Height

Summary The height growth of a hydraulic fracture is known to be affected by many factors that are related to the layered structure of sedimentary rocks. Although these factors are often used to qualitatively explain why hydraulic fractures usually have well–bounded height growth, most of them cannot be directly and quantitatively characterized for a given reservoir to enable a priori prediction of fracture–height growth. In this work, we study the role of the “roughness” of in–situ–stress profiles, in particular alternating low and high stress among rock layers, in determining the tendency of a hydraulic fracture to propagate horizontally vs. vertically. We found that a hydraulic fracture propagates horizontally in low–stress layers ahead of neighboring high–stress layers. Under such a configuration, a fracture–mechanics principle dictates that the net pressure required for horizontal growth of high–stress layers within the current fracture height is significantly lower than that required for additional vertical growth across rock layers. Without explicit consideration of the stress–roughness profile, the system behaves as if the rock is tougher against vertical propagation than it is against horizontal fracture propagation. We developed a simple relationship between the apparent differential rock toughness and characteristics of the stress roughness that induce equivalent overall fracture shapes. This relationship enables existing hydraulic–fracture models to represent the effects of rough in–situ stress on fracture growth without directly representing the fine–resolution rough–stress profiles.

Borehole Cross-Sectional Shape Analysis under in Situ Stress

2020 ◽  
Vol 20 (6) ◽  
pp. 04020045 ◽  
Author(s):  
Zengqiang Han ◽  
Chuanying Wang ◽  
Yiteng Wang ◽  
Chao Wang
Keyword(s):  
Shape Analysis ◽  
In Situ Stress ◽  
Cross Sectional ◽  
Sectional Shape

On the cross-sectional shape of the cellulose crystallites in the tunicate Halocynthia papillosa

1990 ◽  
Vol 48 (3) ◽  
pp. 566-567
Author(s):  
J.-F. Revol ◽  
Y. Van Daele ◽  
F. Gaill
Keyword(s):  
Contrast Imaging ◽  
Animal Kingdom ◽  
Bright Field ◽  
Field Mode ◽  
Cross Sectional ◽  
Ultrathin Sections ◽  
The Cross ◽  
Sectional Shape ◽  
Valonia Ventricosa ◽  
C Nmr

The only form of cellulose which could unequivocally be ascribed to the animal kingdom is the tunicin that occurs in the tests of the tunicates. Recently, high-resolution solid-state l3C NMR revealed that tunicin belongs to the Iβ form of cellulose as opposed to the Iα form found in Valonia and bacterial celluloses. The high perfection of the tunicin crystallites led us to study its crosssectional shape and to compare it with the shape of those in Valonia ventricosa (V.v.), the goal being to relate the cross-section of cellulose crystallites with the two allomorphs Iα and Iβ.In the present work the source of tunicin was the test of the ascidian Halocvnthia papillosa (H.p.). Diffraction contrast imaging in the bright field mode was applied on ultrathin sections of the V.v. cell wall and H.p. test with cellulose crystallites perpendicular to the plane of the sections. The electron microscope, a Philips 400T, was operated at 120 kV in a low intensity beam condition.

scholarly journals Optimization of the cross-sectional shape of the belts of trihedral lattice supports

2019 ◽  
Vol 7 (4) ◽  
pp. 5-8
Author(s):  
Linar Sabitov ◽  
Ilnar Baderddinov ◽  
Anton Chepurnenko
Keyword(s):  
Objective Function ◽  
Nonlinear Optimization ◽  
Cross Section ◽  
Optimization Methods ◽  
Cross Sectional ◽  
Maximum Value ◽  
Axial Moment ◽  
The Cross ◽  
Nonlinear Optimization Methods ◽  
Sectional Shape

The article considers the problem of optimizing the geometric parameters of the cross section of the belts of a trihedral lattice support in the shape of a pentagon. The axial moment of inertia is taken as the objective function. Relations are found between the dimensions of the pentagonal cross section at which the objective function takes the maximum value. We introduce restrictions on the constancy of the consumption of material, as well as the condition of equal stability. The solution is performed using nonlinear optimization methods in the Matlab environment.

Numerical study for the improvement of bead spreading architecture with modified nozzle geometries in additive manufacturing of polymers

2021 ◽  
Vol ahead-of-print (ahead-of-print) ◽  
Author(s):  
Easir Arafat Papon ◽  
Anwarul Haque ◽  
Muhammad Ali Rob Sharif
Keyword(s):  
Free Surface ◽  
Numerical Model ◽  
3D Printing ◽  
Cross Section ◽  
Nozzle Exit ◽  
Cross Sectional ◽  
Content Type ◽  
The Cross ◽  
Viscous Polymer ◽  
Sectional Shape

Purpose This paper aims to develop a numerical model of bead spreading architecture of a viscous polymer in fused filament fabrication (FFF) process with different nozzle geometry. This paper also focuses on the manufacturing feasibility of the nozzles and 3D printing of the molten beads using the developed nozzles. Design/methodology/approach The flow of a highly viscous polymer from a nozzle, the melt expansion in free space and the deposition of the melt on a moving platform are captured using the FLUENT volume of fluid (VOF) method based computational fluid dynamics code. The free surface motion of the material is captured in VOF, which is governed by the hydrodynamics of the two-phase flow. The phases involved in the numerical model are liquid polymer and air. A laminar, non-Newtonian and non-isothermal flow is assumed. Under such assumptions, the spreading characteristic of the polymer is simulated with different nozzle-exit geometries. The governing equations are solved on a regular stationary grid following a transient algorithm, where the boundary between the polymer and the air is tracked by piecewise linear interface construction (PLIC) to reconstruct the free surface. The prototype nozzles were also manufactured, and the deposition of the molten beads on a flatbed was performed using a commercial 3D printer. The deposited bead cross-sections were examined through optical microscopic examination, and the cross-sectional profiles were compared with those obtained in the numerical simulations. Findings The numerical model successfully predicted the spreading characteristics and the cross-sectional shape of the extruded bead. The cross-sectional shape of the bead varied from elliptical (with circular nozzle) to trapezoidal (with square and star nozzles) where the top and bottom surfaces are significantly flattened (which is desirable to reduce the void spaces in the cross-section). The numerical model yielded a good approximation of the bead cross-section, capturing most of the geometric features of the bead with a reasonable qualitative agreement compared to the experiment. The quantitative comparison of the cross-sectional profiles against experimental observation also indicated a favorable agreement. The significant improvement observed in the bead cross-section with the square and star nozzles is the flattening of the surfaces. Originality/value The developed numerical algorithm attempts to address the fundamental challenge of voids and bonding in the FFF process. It presents a new approach to increase the inter-bead bonding and reduce the inter-bead voids in 3D printing of polymers by modifying the bead cross-sectional shape through the modification of nozzle exit-geometry. The change in bead cross-sectional shape from elliptical (circular) to trapezoidal (square and star) cross-section is supposed to increase the contact surface area and inter-bead bonding while in contact with adjacent beads.

Optimization of In-Stage Diversion To Promote Uniform Planar Multifracture Propagation: A Numerical Study

SPE Journal ◽  
2020 ◽  
Vol 25 (06) ◽  
pp. 3091-3110
Author(s):  
Ming Chen ◽  
Shicheng Zhang ◽  
Tong Zhou ◽  
Xinfang Ma ◽  
Yushi Zou
Keyword(s):  
Numerical Study ◽  
High Stress ◽  
In Situ Stress ◽  
Stress Difference ◽  
Multiple Fractures ◽  
Stress Shadow ◽  
Limited Entry ◽  
Fully Coupled ◽  
Flux Redistribution

Summary Creating uniform multiple fractures is a challenging task due to reservoir heterogeneity and stress shadow. Limited-entry perforation and in-stage diversion are commonly used to improve multifracture treatments. Many studies have investigated the mechanism of limited-entry perforation for multifracture treatments, but relatively few have focused on the in-stage diversion process. The design of in-stage diversion is usually through trial and error because of the lack of a simulator. In this study, we present a fully coupled planar 2D multifracture model for simulating the in-stage diversion process. The objective is to evaluate flux redistribution after diversion and optimize the dosage of diverters and diversion timing under different in-stage in-situ stress difference. Our model considers ball sealer allocation and solves flux redistribution after diversion through a fully coupled multifracture model. A supertimestepping explicit algorithm is adopted to solve the solid/fluid coupling equations efficiently. Multifracture fronts are captured by using tip asymptotes and an adaptive time-marching approach. The modeling results are validated against analytical solutions for a plane-strain Khristianovic-Geertsma de Klerk (KGD) model. A series of numerical simulations are conducted to investigate the multifracture growth under different in-stage diversion operations. Parametric studies reveal that the in-stage in-situ stress difference is a critical parameter for diversion designs. When the in-situ stress difference is larger than 2 MPa, the fracture in the high-stress zone can hardly be initiated before diversion for a general fracturing design. More ball sealers are required for the formations with higher in-stage in-situ stress difference. The diverting time should be earlier for formations with high in-stage stress differences as well. Adding more perforation holes in the zone with higher in-situ stress is recommended to achieve even flux distribution. The results of this study can help understand the multifracture growth mechanism during in-stage diversion and optimize the diversion design timely.

scholarly journals Analytical Model of a Multi-Step Straightening Process for Linear Guideways Considering Neutral Axis Deviation

Symmetry ◽  
2018 ◽  
Vol 10 (8) ◽  
pp. 316 ◽  
Author(s):  
Yongquan Zhang ◽  
Hong Lu ◽  
He Ling ◽  
Yang Lian ◽  
Mingtian Ma
Keyword(s):  
Analytical Model ◽  
Cross Section ◽  
Predictive Accuracy ◽  
Neutral Axis ◽  
Longitudinal Stress ◽  
Linear Superposition ◽  
Element Analysis ◽  
Cross Sectional ◽  
The Cross ◽  
Sectional Shape

The cross-sectional shape of a linear guideway has been processed before the straightening process. The cross-section features influence not only the position of the neutral axis, but also the applied and residual stresses along the longitudinal direction, especially in a multi-step straightening process. This paper aims to present an analytical model based on elasto-plastic theory and three-point reverse bending theory to predict straightening stroke and longitudinal stress distribution during the multi-step straightening process of linear guideways. The deviation of the neutral axis is first analyzed considering the asymmetrical features of the cross-section. Owing to the cyclic loading during the multi-step straightening process, the longitudinal stress curves are then calculated using the linear superposition of stresses. Based on the cross-section features and the superposition of stresses, the bending moment is corrected to improve the predictive accuracy of the multi-step straightening process. Finite element analysis, as well as straightening experiments, have been performed to verify the applicability of the analytical model. The proposed approach can be implemented in the multi-step straightening process of linear guideways with similar cross-sectional shape to improve the straightening accuracy.

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